Commit 269a8536 authored by Philipp Arras's avatar Philipp Arras
Browse files

Merge branch 'sebastian_comments' into 'master'

corrected typos

See merge request !2
parents 09de3f11 9b2b5b6c
Pipeline #79996 passed with stage
in 2 minutes and 51 seconds
......@@ -67,7 +67,7 @@ building blocks.
the data, the statistics of the data including the error bars and a
description of the measurement device itself.
\item The \emph{prior} $\mathcal P(s)$ describes the knowledge the scientist has
\emph{before} executing the experiment. In order to define prior one needs to
\emph{before} executing the experiment. In order to define a prior one needs to
make one's knowledge about the physical process which was observed by the
measurement device explicit.
\item Finally, one needs an algorithmic and computational framework which is
......@@ -83,8 +83,8 @@ The IFT group led by Torsten En{\ss}lin has knowledge and GPL-licensed software
to address the third part.
The interesting part is the second one, i.e. thinking about the priors. Here one
needs to have an understanding both of the underlying physics and probability
theory and Bayesian statistics. This is where an interesting interaction between
needs to have an understanding both of the underlying physics and Bayesian probability
theory. This is where an interesting interaction between
observers, i.e. you, and information field theorists, i.e. Torsten and his
group, will happen.
......@@ -125,7 +125,7 @@ If the likelihood is Gaussian
\begin{align*}
\mathcal H(d|s) \propto (d-R(s))^\dagger N^{-1}(d-R(s))
\end{align*}
or Poissonian
with noise covariance $N$ and response $R$,\\ or Poissonian
\begin{align*}
\mathcal H(d|s) \propto - \log (R(s))^\dagger d+\sum_i R(s)_i,
\end{align*}
......@@ -151,11 +151,11 @@ the course of the algorithm. In \texttt{s}, NIFTy will store the reconstruction
of the physical field which the user wants to infer. \texttt{s} would be a
one-dimensional array for a time-series, two-dimensional for a multi-frequency
time-series or for a single-frequency image of the sky, three-dimensional for a
multi-frequency image of sky, etc. To cut a long story short: be aware of the
multi-frequency image of the sky, etc. To cut a long story short: be aware of the
shape of \texttt{s} and \texttt{d}!
\subsection*{The response}
When we talk about \enquote{the response} we mean a function \texttt{R(s)} which
When we talk about \enquote{the response}, we mean a function \texttt{R(s)} which
takes an array of shape \texttt{s.shape} and returns an array of shape
\texttt{d.shape}. This function shall be made such that it simulates the
measurement device. Assume one knows the signal \texttt{s}, what would be the
......@@ -175,7 +175,7 @@ response_out = R(np.ones(shp))
if response_out.shape == d.shape:
print('Yay!')
else:
raise ValueError('Output of response doesn't have the correct shape.')
raise ValueError('Output of response does not have the correct shape.')
\end{lstlisting}
\subsection*{Derivative of response}
......@@ -189,7 +189,7 @@ following shape: \footnote{There are various ways to think about derivatives and
gradients of multi-dimensional functions. A different view on gradients would
be that at a given point $s=\text{position}$ the gradient is a matrix with
\texttt{s.size} columns and \texttt{d.size} rows. Obviously, it is not
feasible to store such a matrix on a computer due to its size. There we think
feasible to store such a matrix on a computer due to its size. Therefore we think
of this matrix in terms of a linear map which maps an array of shape
\texttt{s.shape} to an array of shape \texttt{d.shape}. This linear map shall
be implemented on the computer in terms of a function. Think of this map as a
......@@ -200,7 +200,7 @@ following shape: \footnote{There are various ways to think about derivatives and
What needs to be implemented is a function \texttt{R\_prime(position, s0)} which
takes the arguments \texttt{position} (which is an array of shape \texttt{s.shape}
and determines the position at which we want to calculate the derivative) and
the array \texttt{s0} which shall be the derivative taken of.
the array \texttt{s0} of which the derivative shall be taken.
\texttt{R\_prime} is nonlinear in \texttt{position} in general and linear in
\texttt{s0}. The output of \texttt{R\_prime} is of shape \texttt{d.shape}.
......@@ -304,8 +304,8 @@ Why is this already sufficient?
\item The Hamiltonian $\mathcal H$ is given by: $\mathcal H (d|s) = - \log
(R(s))^\dagger d+\sum_i R(s)_i$. Implementing $R$ and stating that the data is
Poissonian determines this form.
\item Since $R$ is a composition of a convolution and a sampling both of which
is a linear operation, $R$ itself is a linear operator.\footnote{I.e. $R(\alpha
\item Since $R$ is a composition of a convolution and a sampling, both of which
are linear operations, $R$ itself is a linear operator.\footnote{I.e. $R(\alpha
s_1+s_2) = \alpha R(s_1) + R(s_2)$.} Thus, $R' = R$ and $R'^\dagger =
R^\dagger$. All in all, we need an implementation for $R$ and $R^\dagger$.
\end{itemize}
......
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